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'''Garibaldi''' is a [[7-limit]] (and higher) temperament of the [[Schismatic family #Garibaldi|schismatic family]]. It is an extension of [[helmholtz]] temperament beyond the 5-limit but with the same simple [[chain of fifths|chain-of-fifths]] structure (so that [[chain-of-fifths notation|standard notation]] may be used). As in helmholtz temperament, [[5/4]] is mapped to the diminished fourth (e.g. C-F♭), and the new mapping specific to garibaldi is that [[7/4]] is mapped to the double diminished octave (e.g. C-Cbb). This makes garibaldi a [[Marvel temperaments|marvel temperament]].  
{{Infobox regtemp
| Title = Garibaldi
| Subgroups = 2.3.5.7, 2.3.5.7.19
| Comma basis = [[225/224]], [[3125/3087]] (7-limit); <br>[[190/189]], [[225/224]], [[361/360]] (2.3.5.7.19)
| Mapping = 1; 1 -8 -14 -3
| Edo join 1 = 41 | Edo join 2 = 53
| Generators = 3/2
| Generators tuning = 702.10
| Optimization method = CWE
| Pergen = (P8, P5)
| MOS scales = [[5L&nbsp;2s]], [[5L&nbsp;7s]], [[12L&nbsp;5s]], [[12L 17s]]
| Odd limit 1 = 9 | Mistuning 1 = 4.33 | Complexity 1 = 17
| Odd limit 2 = 2.3.5.7.19 21 | Mistuning 2 = 4.65 | Complexity 2 = 17
}}
'''Garibaldi''' is a [[7-limit]] (and higher) [[regular temperament|temperament]] of the [[schismatic family #Garibaldi|schismatic family]]. It is an [[extension]] of [[helmholtz (temperament)|helmholtz]] temperament beyond the 5-limit but with the same simple [[chain of fifths|chain-of-fifths]] structure (so that [[chain-of-fifths notation|standard notation]] may be used). The garibaldi temperament tempers together the Pythagorean, syntonic, and archytas commas into a jack-of-all-trades "generic comma", which can be used to reach intervals of 3, 5, and 7. As in helmholtz temperament, [[5/4]] is mapped to the diminished fourth (e.g. C–F♭; a comma-flat major third), and the new mapping specific to garibaldi is that [[7/4]] is mapped to the double-diminished octave (e.g. C–C𝄫; a comma-flat minor seventh). This makes garibaldi a [[marvel temperaments|marvel]] and [[hemifamity temperaments|hemifamity]] temperament. Tuning the fifth a fraction of a cent sharp gives the best tunings.  


Immediate 11-limit extensions include ''cassandra'' (41 &amp; 53), mapping 11/8 to +23 fifths, ''andromeda'' (29 &amp; 41), mapping 11/8 to -18 fifths, and ''helenus'' (53 &amp; 65d), mapping 11/8 to -30 fifths. Garibaldi is most naturally a 2.3.5.7.19 [[subgroup]] temperament due to its immediate availability of [[19/16]] at the minor third (C-Eb).  
Immediate 11-limit extensions include '''cassandra''' ({{nowrap| 41 & 53 }}), mapping 11/8 to +23 fifths, '''andromeda''' ({{nowrap| 29 & 41 }}), mapping 11/8 to −18 fifths, and '''helenus''' ({{nowrap| 53 & 65d }}), mapping 11/8 to −30 fifths. Garibaldi is most naturally a 2.3.5.7.19-[[subgroup]] temperament due to its immediate availability of [[19/16]] at the minor third (C–E♭). This is sometimes known as ''garibaldi nestoria.''


Garibaldi was named in honor of [[Eduardo Sábat-Garibaldi]].  
Garibaldi was named in honor of [[Eduardo Sábat-Garibaldi]], who developed the [[dinarra]], a 53-tone [[microtonal guitar]] in the 1/9-schisma tuning.
 
See [[Schismatic family #Garibaldi]] for technical data.


== Interval chain ==
== Interval chain ==
In the following table, odd harmonics 1–21 are in '''bold'''.  
In the following table, odd harmonics 1–21 and their inverses are in '''bold'''.  


{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
! rowspan="3" | #
! rowspan="3" | #
! rowspan="3" | Cents*
! rowspan="3" | Cents*
! colspan="4" | Approximate Ratios
! colspan="4" | Approximate ratios
|-
|-
! rowspan="2" | 2.3.5.7.19 Subgroup
! rowspan="2" | 2.3.5.7.19 subgroup
! colspan="3" | 13-limit Extension
! colspan="3" | 13-limit extensions
|-
|-
! Cassandra
! Cassandra
Line 22: Line 39:
| 0
| 0
| 0.00
| 0.00
| 1/1
| '''1/1'''
|
|
|
|
Line 28: Line 45:
|-
|-
| 1
| 1
| 702.06
| 702.10
| '''3/2'''
| '''3/2'''
|
|
Line 35: Line 52:
|-
|-
| 2
| 2
| 204.12
| 204.20
| '''9/8'''
| '''9/8'''
|
|
Line 42: Line 59:
|-
|-
| 3
| 3
| 906.18
| 906.30
| 27/16, '''32/19''', 42/25
| 27/16, '''32/19''', 42/25
| 22/13
| 22/13
Line 49: Line 66:
|-
|-
| 4
| 4
| 408.24
| 408.40
| 19/15, 24/19, 63/50, 80/63
| 19/15, 24/19
|
|
| 14/11
| 14/11
Line 56: Line 73:
|-
|-
| 5
| 5
| 1110.29
| 1110.50
| 19/10, 36/19, 40/21
| 19/10, 36/19, 40/21
|
|
Line 63: Line 80:
|-
|-
| 6
| 6
| 612.35
| 612.60
| 10/7
| 10/7
|
|
Line 70: Line 87:
|-
|-
| 7
| 7
| 114.41
| 114.70
| 15/14, '''16/15'''
| 15/14, '''16/15'''
|
|
Line 77: Line 94:
|-
|-
| 8
| 8
| 816.47
| 816.80
| '''8/5'''
| '''8/5'''
|
|
Line 84: Line 101:
|-
|-
| 9
| 9
| 318.53
| 318.90
| 6/5
| 6/5
|
|
Line 91: Line 108:
|-
|-
| 10
| 10
| 1020.59
| 1021.00
| 9/5, 38/21
| 9/5, 38/21
|
|
Line 98: Line 115:
|-
|-
| 11
| 11
| 522.65
| 523.09
| 19/14, 27/20
| 19/14, 27/20
|
|
Line 105: Line 122:
|-
|-
| 12
| 12
| 24.71
| 25.19
| 50/49, 57/56, 64/63, 81/80
| 50/49, 57/56, 64/63, 81/80
|
|
Line 112: Line 129:
|-
|-
| 13
| 13
| 726.77
| 727.29
| '''32/21'''
| '''32/21'''
|
|
Line 119: Line 136:
|-
|-
| 14
| 14
| 228.82
| 229.39
| '''8/7'''
| '''8/7'''
|
|
Line 126: Line 143:
|-
|-
| 15
| 15
| 930.88
| 931.49
| 12/7
| 12/7
|
|
Line 133: Line 150:
|-
|-
| 16
| 16
| 432.94
| 433.59
| 9/7
| 9/7
|
|
Line 140: Line 157:
|-
|-
| 17
| 17
| 1135.00
| 1135.69
| 27/14, 48/25
| 27/14, 48/25
| 52/27
| 52/27
Line 147: Line 164:
|-
|-
| 18
| 18
| 637.06
| 637.79
| 36/25, 81/56
| 36/25, 81/56
| 13/9
| 13/9
Line 154: Line 171:
|-
|-
| 19
| 19
| 139.12
| 139.89
| 27/25
| 27/25
| 13/12
| 13/12
Line 161: Line 178:
|-
|-
| 20
| 20
| 841.18
| 841.99
| 80/49, 81/50
| 57/35, 80/49
| '''13/8''', 44/27
| '''13/8''', 44/27
| 18/11, 64/39
| 18/11, 64/39
Line 168: Line 185:
|-
|-
| 21
| 21
| 343.24
| 344.09
| 60/49
| 60/49
| 11/9, 39/32
| 11/9, 39/32
Line 175: Line 192:
|-
|-
| 22
| 22
| 1045.30
| 1046.19
| 64/35
| 64/35
| 11/6
| 11/6
Line 182: Line 199:
|-
|-
| 23
| 23
| 547.35
| 548.29
| 48/35
| 48/35
| '''11/8''', 26/19
| '''11/8''', 26/19
Line 189: Line 206:
|-
|-
| 24
| 24
| 49.41
| 50.39
| 36/35
| 36/35
| 33/32
| 33/32
Line 196: Line 213:
|-
|-
| 25
| 25
| 751.47
| 752.49
| 54/35
| 54/35
|
|
Line 203: Line 220:
|-
|-
| 26
| 26
| 253.53
| 254.59
| 81/70, 144/125
| 57/49, 81/70, 144/125
| 22/19
| 22/19
|
|
Line 210: Line 227:
|-
|-
| 27
| 27
| 955.59
| 956.69
| 216/125, 256/147
| 171/98, 216/125, 256/147
| 26/15
| 26/15
|
|
Line 217: Line 234:
|-
|-
| 28
| 28
| 457.65
| 458.79
| 64/49
| 64/49
| 13/10
| 13/10
Line 224: Line 241:
|-
|-
| 29
| 29
| 1159.71
| 1160.89
| 96/49
| 96/49
| 39/20, 88/45
| 39/20, 88/45
Line 231: Line 248:
|-
|-
| 30
| 30
| 661.77
| 662.99
| 72/49
| 72/49
| 22/15
| 22/15
Line 238: Line 255:
|-
|-
| 31
| 31
| 163.83
| 165.08
| 54/49
| 54/49
| 11/10
| 11/10
Line 245: Line 262:
|-
|-
| 32
| 32
| 865.88
| 867.18
| 81/49
| 81/49
| 33/20
| 33/20
Line 252: Line 269:
|-
|-
| 33
| 33
| 367.94
| 369.28
| 216/175
| 216/175
| 26/21
| 26/21
Line 259: Line 276:
|-
|-
| 34
| 34
| 1070.00
| 1071.38
| 324/175
| 324/175
| 13/7
| 13/7
Line 266: Line 283:
|-
|-
| 35
| 35
| 572.06
| 573.48
| 243/175
| 243/175
|  
|  
Line 273: Line 290:
|-
|-
| 36
| 36
| 74.12
| 75.58
| 256/245
| 256/245
| 22/21
| 22/21
Line 280: Line 297:
|-
|-
| 37
| 37
| 776.18
| 777.68
| 384/245
| 384/245
| 11/7
| 11/7
Line 287: Line 304:
|-
|-
| 38
| 38
| 278.24
| 279.78
| 288/245
| 288/245
|  
|  
Line 294: Line 311:
|-
|-
| 39
| 39
| 980.30
| 981.88
| 432/245
| 432/245
|  
|  
Line 301: Line 318:
|-
|-
| 40
| 40
| 482.36
| 483.98
| 324/245
| 324/245
|  
|  
Line 308: Line 325:
|-
|-
| 41
| 41
| 1184.41
| 1186.08
| 486/245
| 486/245
|  
|  
Line 314: Line 331:
|  
|  
|}
|}
<nowiki>*</nowiki> in 7-limit CTE tuning
<nowiki/>* In 2.3.5.7.19-subgroup CWE tuning
 
=== As a detemperament of 12et ===
[[File:Garibaldi 12et Detempering.png|thumb|Garibaldi as a 41-tone 12et detempering]]
[[File:Garibaldi-cassandra 12et Detempering.png|thumb|Garibaldi/cassandra as a 53-tone 12et detempering]]
 
Garibaldi is very naturally considered as a [[detemperament]] of the [[12edo|12 equal temperament]] (12et), where the chromatic scale becomes a near-equal [[5L 7s]]. The diagram on the right shows a 53-tone detempered scale, with a generator range of -26 to +26. 53 is the largest number of tones for a mos where the 12 categories never overlap. 
 
Each pitch category of 12et is further divided into four or five qualities, separated by a [[pythagorean comma]], which represents the syntonic~septimal comma. Combining this division with the minor and major diatonic qualities of 12et, garibaldi can give up to ''eight'' qualities for each diatonic category. Taking thirds as an example: 
 
In 12tet:
 
* 7/6~19/16~6/5 (minor)
* 5/4~19/15~9/7 (major)
 
In garibaldi (cassandra)
 
* ~[[7/6]] (subminor)
* '''~[[19/16]] (minor)'''
* ~[[6/5]] (superminor)
* ~[[11/9]] (artoneutral)
* ~[[27/22]] (tendoneutral)
* ~[[5/4]] (submajor)
* '''~[[19/15]] (major)'''
* ~[[9/7]] (supermajor)
 
Notice also the little interval between artoneutral and tendoneutral, ~[[243/242]]. This interval spans 41 generator steps. 41edo tempers it out so that it merges artoneutral and tendoneutral into a [[Sqrt(3/2)|hemififth]] whereas 53edo exaggerates it to the size of the generic comma. 94edo tunes it to one half the size of the general comma, which can be seen as a good compromise.
 
On another note, excluding 41edo, the two neutral intervals also have natural 13-limit interpretations in cassandra: 11/9~[[39/32]] and 27/22~[[16/13]], tempering out [[352/351]]. This also means the minor third is ~[[13/11]].


== Notation ==
== Notation ==
Using garibaldi can be a challenge because it defies the tradition of tertian harmony in [[circle-of-fifths notation]]. The just major triad on C is C-Fb-G, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C-vE-G.  
Like in [[schismic]], it is recommended to adopt an additional module of accidentals such as arrows to represent the comma step. Garibaldi further benefits from this as the arrow also stands in for the septimal comma, so that the same inflection can be used to reach classical and septimal intervals alike.
 
The following table shows how to notate 2.3.5.7.11.13.19 intervals in each extension of garibaldi.  


{| class="wikitable center-1 center-3"
{| class="wikitable" style="text-align:center; vertical-align:middle;"
|+Cassandra nomenclature<br>for selected intervals
|+Nomenclature of selected intervals
! Ratio
|- style="font-weight:bold;"
! Nominal
! rowspan="2" | Ratio
! Example
! colspan="3" | Example
|- style="font-weight:bold;"
| Cassandra
| Andromeda
| Helenus
|-
|-
| 3/2
| 3/2
| Perfect fifth
| colspan="3" | C–G (perfect fifth)
| C-G
|-
|-
| 5/4
| 5/4
| Down major third
| colspan="3" | C–↓E (downmajor third)
| C-vE
|-
|-
| 7/4
| 7/4
| Down minor seventh
| colspan="3" | C–↓Bb (downminor seventh)
| C-vBb
|-
|-
| 11/8
| 11/8
| Double-up fourth
| C–↑↑F (dupfourth)
| C-^^F
| C–↓↓F#* (dudtritone)
| C–↓3F#* (trudtritone)
|-
|-
| 13/8
| 13/8
| Double-up minor sixth
| C–↑↑Ab (dupminor sixth)
| C-^^Ab
| C–↓↓A (dudmajor sixth)
| C–↓3A (trudmajor sixth)
|-
|-
| 19/16
| 19/16
| Minor third
| colspan="3" | C–Eb (minor third)
| C-Eb
|}
|}


{| class="wikitable center-1 center-3 mw-collapsible mw-collapsed"
<nowiki/>*Can also be spelt ↓Gb and ↓↓Gb respectively, since F# = ↑Gb.
|+ style=white-space:nowrap | Andromeda nomenclature for selected intervals
! Ratio
! Nominal
! Example
|-
| 11/8
| Down diminished fifth<br>Double-down augmented fourth
| C-vGb<br>C-vvF#
|-
| 13/8
| Double down major sixth
| C-vvA
|}
 
{| class="wikitable center-1 center-3 mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | Helenus nomenclature for selected intervals
! Ratio
! Nominal
! Example
|-
| 11/8
| Double-down diminished fifth<br>Triple-down augmented fourth
| C-vvGb<br>C-v<sup>3</sup>F#
|-
| 13/8
| Triple-down major sixth
| C-v<sup>3</sup>A
|}


== Chords and harmony ==
== Chords and harmony ==
Traditional tertian harmony is effective. The default triads on the Pythagorean spine are undevicesimal in quality:  
Traditional tertian harmony is effective. The default triads on the Pythagorean spine are undevicesimal in quality:  
* 1-19/15-3/2 (C-E-G)
* 1–19/15–3/2 (C–E–G)
* 1-19/16-3/2 (C-Eb-G)
* 1–19/16–3/2 (C–Eb–G)


Note that the major third also represents [[24/19]], and the minor third, [[13/11]]. These chords are typically associated with a sort of coldness and metalness, like those in [[12edo]] if not more so.  
Note that the major third also represents [[24/19]], and the minor third, [[13/11]]. These chords are typically associated with a sort of coldness and metalness, like those in [[12edo]] if not more so.  


If a warm, sweet, laid-back sound is desired, the thirds can be inflected inwards by a comma to yield
If a warm, sweet, laid-back sound is desired, the thirds can be inflected inwards by a comma to yield
* 1-5/4-3/2 (C-vE-G)
* 1–5/4–3/2 (C–↓E–G)
* 1-6/5-3/2 (C-^Eb-G)
* 1–6/5–3/2 (C–↑Eb–G)


Contrarily, for a more sour and active sound, they can be inflected outwards by a comma to yield
Contrarily, for a more sour and active sound, they can be inflected outwards by a comma to yield
* 1-9/7-3/2 (C-^E-G)
* 1–9/7–3/2 (C–↑E-G)
* 1-7/6-3/2 (C-vEb-G)
* 1–7/6–3/2 (C–↓Eb-G)


== Scales ==
== Scales ==
Line 403: Line 424:


== Tunings ==
== Tunings ==
=== Tuning spectra ===
=== Norm-based tunings ===
==== Cassandra ====
{| class="wikitable mw-collapsible mw-collapsed"
Gencom: [2 4/3; 225/224 275/273 325/324 385/384]
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 702.0589{{c}}
| CWE: ~3/2 = 702.0774{{c}}
| POTE: ~3/2 = 702.0852{{c}}
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings (cassandra)
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 702.1192{{c}}
| CWE: ~3/2 = 702.1135{{c}}
| POTE: ~3/2 = 702.1125{{c}}
|}
 
=== Target tunings ===
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (garibaldi)
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 7-odd-limit
| ~3/2 = 702.2086{{c}}
| 7/6
| ~3/2 = 702.140{{c}}
| {{Monzo| 0 -25 11 35 }}
|-
| 9-odd-limit
| ~3/2 = 702.1928{{c}}
| 9/7
| ~3/2 = 702.114{{c}}
| {{Monzo| 0 -27 7 17 }}
|}
 
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (cassandra)
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 11-odd-limit
| ~3/2 = 702.1928{{c}}
| 9/7
| ~3/2 = 702.183{{c}}
| {{Monzo| 0 17 -52 -88 134 }}
|-
| 13-odd-limit
| ~3/2 = 702.1089{{c}}
| 13/7
| ~3/2 = 702.128{{c}}
| {{Monzo| 0 -38 -80 -122 137 116 }}
|-
| 15-odd-limit
| ~3/2 = 702.1089{{c}}
| 13/7
| ~3/2 = 702.112{{c}}
| {{Monzo| 0 -95 -137 -129 167 143 }}
|}


Gencom mapping: {{mapping| 1 2 -1 -3 13 12 | 0 -1 8 14 -23 -20 }}
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (andromeda)
! rowspan="2" | Target
! colspan="2" | Minimax
|-
! Generator
! Eigenmonzo*
|-
| 11-odd-limit
| ~3/2 = 702.6296{{c}}
| 11/9
|-
| 13-odd-limit
| ~3/2 = 702.7558{{c}}
| 13/9
|-
| 15-odd-limit
| ~3/2 = 702.7558{{c}}
| 13/9
|}


{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (helenus)
! rowspan="2" | Target
! colspan="2" | Minimax
|-
! Generator
! Eigenmonzo*
|-
| 11-odd-limit
| ~3/2 = 701.6435{{c}}
| 11/9
|-
| 13-odd-limit
| ~3/2 = 701.6435{{c}}
| 11/9
|-
| 15-odd-limit
| ~3/2 = 701.6435{{c}}
| 11/9
|}
=== Tuning spectra ===
==== Garibaldi ====
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
| '''[[12edo|7\12]]'''
|
| '''700.0000'''
| '''Lower bound of 9-odd-limit, <br>2.3.5.7.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|
| 19/16
| 700.8290
| 1/3 undevicesimal schisma
|-
|-
! Edo<br>Generator
|
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
| 19/12
| 701.1105
| 1/4 undevicesimal schisma
|-
| [[65edo|38\65]]
|
| 701.5385
| 65d val
|-
|
| 15/8
| 701.6759
| 1/7 schisma
|-
|
| 5/4
| 701.7108
| 1/8 schisma
|-
|
| 25/24
| 701.7252
| 2/17 schisma
|-
|
| 5/3
| 701.7379
| 5-odd-limit minimax, 1/9 schisma
|-
|
| 9/5
| 701.7596
| 1/10 schisma
|-
|
| 81/80
| 701.7922
| 1/12 schisma
|-
| [[53edo|31\53]]
|
| 701.8868
|
|-
|
| 3/2
| 701.9550
| Pythagorean tuning
|-
|
| 36/35
| 702.0321
|
|-
| [[94edo|55\94]]
|
| 702.1277
|
|-
|
| 9/7
| 702.1928
| 9-odd-limit minimax, 1/16 septimal schisma
|-
|
| 7/6
| 702.2086
| 7-odd-limit minimax, 1/15 septimal schisma
|-
|
| 49/48
| 702.2174
| 2/29 septimal schisma
|-
|
| 7/4
| 702.2267
| 1/14 septimal schisma
|-
|
| 19/10
| 702.2399
|
|-
|
| 21/16
| 702.2476
| 1/13 septimal schisma
|-
|
| 64/63
| 702.2720
| 1/12 septimal schisma
|-
|
| 19/15
| 702.3111
|
|-
| [[41edo|24\41]]
|
| 702.4390
|
|-
|
| 19/14
| 702.6079
|
|-
|
| 21/19
| 702.6732
|
|-
|
| 15/14
| 702.7775
|
|-
|
| 7/5
| 702.9146
|
|-
|
| 21/20
| 703.1066
|
|-
| '''[[29edo|17\29]]'''
|
| '''703.4483'''
| '''Upper bound of 9-odd-limit, <br>2.3.5.7.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|
| 13/11
| 703.5968
|
|}
 
==== Cassandra ====
{| class="wikitable mw-collapsible mw-collapsed center-all left-4"
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
| 7\12
| '''[[12edo|7\12]]'''
|  
|  
| 700.0000
| '''700.0000'''
| Lower bound of 9-odd-limit diamond monotone
| '''Lower bound of 9-odd-limit diamond monotone'''
|-
|-
|  
|  
Line 431: Line 735:
| 1/4 undevicesimal schisma
| 1/4 undevicesimal schisma
|-
|-
| 38\65
| [[65edo|38\65]]
|  
|  
| 701.5385
| 701.5385
|  
| 65def val
|-
|-
|  
|  
| 15/8
| 15/8
| 701.676
| 701.6759
| 1/7 schisma
| 1/7 schisma
|-
|-
|  
|  
| 5/4
| 5/4
| 701.711
| 701.7108
| 1/8 schisma
| 1/8 schisma
|-
|-
|  
|  
| {{monzo| 0 -10 17 }}
| 25/24
| 701.728
| 701.7252
| 5-odd-limit least squares
| 2/17 schisma
|-
|-
|  
|  
| 5/3
| 5/3
| 701.738
| 701.7379
| 5-odd-limit minimax, 1/9 schisma
| 5-odd-limit minimax, 1/9 schisma
|-
|-
|  
|  
| 9/5
| 9/5
| 701.760
| 701.7596
| 1/10 schisma
| 1/10 schisma
|-
|
| 81/80
| 701.7922
| 1/12 schisma
|-
|-
|  
|  
Line 466: Line 775:
|  
|  
|-
|-
| 31\53
| '''[[53edo|31\53]]'''
|  
|  
| 701.8868
| '''701.8868'''
| Lower bound of 11-, 13-, and 15-odd-limit diamond monotone
| '''Lower bound of 11-, 13-, 15-odd-limit, <br>2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|-
|  
|  
Line 488: Line 797:
|  
|  
| 13/8
| 13/8
| 702.026
| 702.0264
|  
|  
|-
|-
|  
|  
| 13/12
| 13/12
| 702.030
| 702.0301
|
|-
|
| 36/35
| 702.0321
|  
|  
|-
|-
|  
|  
| 13/9
| 13/9
| 702.034
| 702.0343
|  
|  
|-
|-
Line 508: Line 822:
|  
|  
| 11/10
| 11/10
| 702.097
| 702.0969
|  
|  
|-
|-
|  
|  
| 15/11
| 15/11
| 702.102
| 702.1016
|  
|  
|-
|-
|  
|  
| 13/7
| 13/7
| 702.109
| 702.1089
| 13- and 15-odd-limit minimax
| 13- and 15-odd-limit minimax
|-
|
| <span style="font-size:0.75em">{{monzo| 0 -95 -137 -129 167 143 }}</span>
| 702.112
| 15-odd-limit least squares
|-
|-
|  
|  
Line 531: Line 840:
|  
|  
|-
|-
| [[94edo|55\94]]
|  
|  
| {{monzo| 0 -27 7 17 }}
| 702.1277
| 702.114
| 9-odd-limit least squares
|-
|
| <span style="font-size:0.75em">{{monzo| 0 -38 -80 -122 137 116 }}</span>
| 702.128
| 13-odd-limit least squares
|-
|  
|  
| {{monzo| 0 -25 11 35 }}
| 702.140
| 7-odd-limit least squares
|-
|
| <span style="font-size:0.9em">{{monzo| 0 17 -52 -88 134 }}</span>
| 702.183
| 11-odd-limit least squares
|-
|-
|  
|  
| 9/7
| 9/7
| 702.193
| 702.1928
| 9- and 11-odd-limit minimax, 1/16 septimal schisma
| 9- and 11-odd-limit minimax, 1/16 septimal schisma
|-
|-
|  
|  
| 7/6
| 7/6
| 702.209
| 702.2086
| 7-odd-limit minimax, 1/15 septimal schisma
| 7-odd-limit minimax, 1/15 septimal schisma
|-
|
| 49/48
| 702.2174
| 2/29 septimal schisma
|-
|-
|  
|  
| 7/4
| 7/4
| 702.227
| 702.2267
| 1/14 septimal schisma
| 1/14 septimal schisma
|-
|-
|  
|  
| 11/7
| 11/7
| 702.230
| 702.2295
|  
|  
|-
|-
|  
|  
| 11/8
| 11/8
| 702.231
| 702.2312
|  
|  
|-
|-
Line 588: Line 887:
|  
|  
| 11/6
| 11/6
| 702.244
| 702.2438
|  
|  
|-
|-
Line 598: Line 897:
|  
|  
| 11/9
| 11/9
| 702.258
| 702.2575
|
|-
|  
|  
| 64/63
| 702.2720
| 1/12 septimal schisma
|-
|-
|  
|  
Line 606: Line 910:
|  
|  
|-
|-
| 24\41
| '''[[41edo|24\41]]'''
|  
|  
| 702.4390
| '''702.4390'''
| Upper bound of 11-, 13-, and 15-odd-limit diamond monotone
| '''Upper bound of 11-, 13-, 15-odd-limit, <br>2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|-
|  
|  
Line 623: Line 927:
|  
|  
| 15/14
| 15/14
| 702.778
| 702.7775
|  
|  
|-
|-
|  
|  
| 7/5
| 7/5
| 702.915
| 702.9146
|  
|  
|-
|-
Line 636: Line 940:
|  
|  
|-
|-
| 17\29
| '''[[29edo|17\29]]'''
|  
|  
| 703.4483
| '''703.4483'''
| Upper bound of 9-odd-limit diamond monotone
| '''29ef val, upper bound of 9-odd-limit diamond monotone'''
|-
|-
|  
|  
| 13/11
| 13/11
| 703.597
| 703.5968
|  
|  
|}
|}


==== Andromeda ====
==== Andromeda ====
Gencom: [2 4/3; 100/99 105/104 196/195 245/242]
{| class="wikitable mw-collapsible mw-collapsed center-all left-4"
 
! Edo<br>generator
Gencom mapping: {{mapping| 1 2 -1 -3 -4 -5 | 0 -1 8 14 18 21 }}
! Unchanged interval<br>(eigenmonzo)*
 
{| class="wikitable center-all left-4"
|-
! Edo<br>Generator
! Eigenmonzo<br>(Unchanged-interval)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
| 7\12
| '''[[12edo|7\12]]'''
|  
|  
| 700.0000
| '''700.0000'''
| Lower bound of 9- and 11-odd-limit diamond monotone
| '''Lower bound of 9- and 11-odd-limit diamond monotone'''
|-
|-
|  
|  
Line 674: Line 973:
| 1/4 undevicesimal schisma
| 1/4 undevicesimal schisma
|-
|-
| 38\65
| [[65edo|38\65]]
|  
|  
| 701.5385
| 701.5385
|  
| 65deeff val
|-
|-
|  
|  
| 15/8
| 15/8
| 701.676
| 701.6759
| 1/7 schisma
| 1/7 schisma
|-
|-
|  
|  
| 5/4
| 5/4
| 701.711
| 701.7108
| 1/8 schisma
| 1/8 schisma
|-
|
| 25/24
| 701.7252
| 2/17 schisma
|-
|-
|  
|  
| 5/3
| 5/3
| 701.738
| 701.7379
| 5-odd-limit minimax, 1/9 schisma
| 5-odd-limit minimax, 1/9 schisma
|-
|-
|  
|  
| 9/5
| 9/5
| 701.760
| 701.7596
| 1/10 schisma
| 1/10 schisma
|-
|-
| 31\53
|
| 81/80
| 701.7922
| 1/12 schisma
|-
| [[53edo|31\53]]
|  
|  
| 701.8868
| 701.8868
|  
| 53ef val
|-
|-
|  
|  
Line 708: Line 1,017:
| 701.9550
| 701.9550
| Pythagorean tuning
| Pythagorean tuning
|-
|
| 36/35
| 702.0321
|
|-
|-
|  
|  
| 9/7
| 9/7
| 702.193
| 702.1928
| 9-odd-limit minimax, 1/16 septimal schisma
| 9-odd-limit minimax, 1/16 septimal schisma
|-
|-
|  
|  
| 7/6
| 7/6
| 702.209
| 702.2086
| 7-odd-limit minimax, 1/15 septimal schisma
| 7-odd-limit minimax, 1/15 septimal schisma
|-
|
| 49/48
| 702.2174
| 2/29 septimal schisma
|-
|-
|  
|  
| 7/4
| 7/4
| 702.227
| 702.2267
| 1/14 septimal schisma
| 1/14 septimal schisma
|-
|-
Line 728: Line 1,047:
| 702.2476
| 702.2476
| 1/13 septimal schisma
| 1/13 septimal schisma
|-
|
| 64/63
| 702.2720
| 1/12 septimal schisma
|-
|-
|  
|  
Line 734: Line 1,058:
|  
|  
|-
|-
| 24\41
| '''[[41edo|24\41]]'''
|  
|  
| 702.4390
| '''702.4390'''
| Lower bound of 13- and 15-odd-limit diamond monotone
| '''Lower bound of 13-, 15-odd-limit, <br>2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|-
|  
|  
Line 746: Line 1,070:
|  
|  
| 11/9
| 11/9
| 702.630
| 702.6296
| 11-odd-limit minimax
| 11-odd-limit minimax
|-
|-
|  
|  
| 11/6
| 11/6
| 702.665
| 702.6651
|  
|  
|-
|-
Line 761: Line 1,085:
|  
|  
| 11/8
| 11/8
| 702.705
| 702.7046
|  
|  
|-
|-
|  
|  
| 13/9
| 13/9
| 702.756
| 702.7558
| 13- and 15-odd-limit minimax
| 13- and 15-odd-limit minimax
|-
|-
|  
|  
| 15/14
| 15/14
| 702.778
| 702.7775
|  
|  
|-
|-
|  
|  
| 13/12
| 13/12
| 702.792
| 702.7922
|  
|  
|-
|-
|  
|  
| 13/8
| 13/8
| 702.832
| 702.8320
|  
|  
|-
|-
|  
|  
| 7/5
| 7/5
| 702.915
| 702.9146
|  
|  
|-
|-
Line 806: Line 1,130:
|  
|  
| 15/11
| 15/11
| 703.359
| 703.3592
|  
|  
|-
|-
|  
|  
| 15/13
| 15/13
| 703.410
| 703.4101
|  
|  
|-
|-
| 17\29
| '''[[29edo|17\29]]'''
|  
|  
| 703.4483
| '''703.4483'''
| Upper bound of 9-, 11, 13, and 15-odd-limit diamond monotone
| '''Upper bound of 9-, 11-, 13-, 15-odd-limit, <br>2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|-
|  
|  
| 11/10
| 11/10
| 703.500
| 703.4996
|  
|  
|-
|-
|  
|  
| 13/10
| 13/10
| 703.522
| 703.5220
|  
|  
|-
|-
|  
|  
| 13/11
| 13/11
| 703.597
| 703.5968
|  
|  
|-
|-
Line 851: Line 1,175:
|  
|  
| 13/7
| 13/7
| 704.043
| 704.0426
|  
|  
|-
|-
|  
|  
| 11/7
| 11/7
| 704.377
| 704.3770
|  
|  
|}
|}


==== Helenus ====
==== Helenus ====
Gencom: [2 4/3; 99/98 176/175 275/273 847/845]
{| class="wikitable mw-collapsible mw-collapsed center-all left-4"
 
! Edo<br>generator
Gencom mapping: {{mapping| 1 2 -1 -3 -9 -10 | 0 -1 8 14 30 33 }}
! Unchanged interval<br>(eigenmonzo)*
 
{| class="wikitable center-all left-4"
|-
! Edo<br>Generator
! Eigenmonzo<br>(Unchanged-interval)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
| 7\12
| '''[[12edo|7\12]]'''
|  
|  
| 700.0000
| '''700.0000'''
| Lower bound of 9- and 11-odd-limit diamond monotone
| '''Lower bound of 9- and 11-odd-limit diamond monotone'''
|-
|-
|  
|  
Line 884: Line 1,203:
|  
|  
| 11/7
| 11/7
| 701.094
| 701.0942
|  
|  
|-
|-
Line 899: Line 1,218:
|  
|  
| 13/7
| 13/7
| 701.489
| 701.4894
|  
|  
|-
|-
Line 907: Line 1,226:
|  
|  
|-
|-
| 38\65
| '''[[65edo|38\65]]'''
|  
|  
| 701.5385
| '''701.5385'''
| Lower bound of 13- and and 15-odd-limit diamond monotone
| '''65d val, lower bound of 13-, 15-odd-limit, <br>2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|-
|  
|  
| 11/10
| 11/10
| 701.591
| 701.5907
|  
|  
|-
|-
|  
|  
| 15/11
| 15/11
| 701.607
| 701.6066
|  
|  
|-
|-
|  
|  
| 11/8
| 11/8
| 701.623
| 701.6227
|  
|  
|-
|-
|  
|  
| 11/6
| 11/6
| 701.633
| 701.6335
|  
|  
|-
|-
|  
|  
| 11/9
| 11/9
| 701.644
| 701.6435
| 11-, 13-, and 15-odd-limit minimax
| 11-, 13-, and 15-odd-limit minimax
|-
|-
|  
|  
| 15/8
| 15/8
| 701.676
| 701.6759
| 1/7 schisma
| 1/7 schisma
|-
|-
Line 949: Line 1,268:
|  
|  
| 5/4
| 5/4
| 701.711
| 701.7108
| 1/8 schisma
| 1/8 schisma
|-
|
| 25/24
| 701.7252
| 2/17 schisma
|-
|-
|  
|  
| 5/3
| 5/3
| 701.738
| 701.7379
| 5-odd-limit minimax, 1/9 schisma
| 5-odd-limit minimax, 1/9 schisma
|-
|-
|  
|  
| 9/5
| 9/5
| 701.760
| 701.7596
| 1/10 schisma
| 1/10 schisma
|-
|
| 81/80
| 701.7922
| 1/12 schisma
|-
|-
|  
|  
| 13/8
| 13/8
| 701.802
| 701.8022
|  
|  
|-
|-
|  
|  
| 13/12
| 13/12
| 701.807
| 701.8067
|  
|  
|-
|-
|  
|  
| 13/9
| 13/9
| 701.811
| 701.8109
|  
|  
|-
|-
|  
|  
| 13/10
| 13/10
| 701.831
| 701.8314
|  
|  
|-
|-
|  
|  
| 15/13
| 15/13
| 701.836
| 701.8362
|  
|  
|-
|-
| 31\53
| '''[[53edo|31\53]]'''
|  
|  
| 701.8868
| '''701.8868'''
| Upper bound of 11-, 13-, and 15-odd-limit diamond monotone
| '''Upper bound of 11-, 13-, 15-odd-limit, <br>2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|-
|  
|  
Line 1,001: Line 1,330:
| 701.9550
| 701.9550
| Pythagorean tuning
| Pythagorean tuning
|-
|
| 36/35
| 702.0321
|
|-
|-
|  
|  
| 9/7
| 9/7
| 702.193
| 702.1928
| 9-odd-limit minimax, 1/16 septimal schisma
| 9-odd-limit minimax, 1/16 septimal schisma
|-
|-
|  
|  
| 7/6
| 7/6
| 702.209
| 702.2086
| 7-odd-limit minimax, 1/15 septimal schisma
| 7-odd-limit minimax, 1/15 septimal schisma
|-
|
| 49/48
| 702.2174
| 2/29 septimal schisma
|-
|-
|  
|  
| 7/4
| 7/4
| 702.227
| 702.2267
| 1/14 septimal schisma
| 1/14 septimal schisma
|-
|-
Line 1,026: Line 1,365:
| 702.2476
| 702.2476
| 1/13 septimal schisma
| 1/13 septimal schisma
|-
|
| 64/63
| 702.2720
| 1/12 septimal schisma
|-
|-
|  
|  
Line 1,032: Line 1,376:
|  
|  
|-
|-
| 24\41
| [[41edo|24\41]]
|  
|  
| 702.4390
| 702.4390
|  
| 41ef val
|-
|-
|  
|  
Line 1,049: Line 1,393:
|  
|  
| 15/14
| 15/14
| 702.778
| 702.7775
|  
|  
|-
|-
|  
|  
| 7/5
| 7/5
| 702.915
| 702.9146
|  
|  
|-
|-
Line 1,062: Line 1,406:
|  
|  
|-
|-
| 17\29
| '''[[29edo|17\29]]'''
|  
|  
| 703.4483
| '''703.4483'''
| Upper bound of 9-odd-limit diamond monotone
| '''29eeff val, upper bound of 9-odd-limit diamond monotone'''
|-
|-
|  
|  
| 13/11
| 13/11
| 703.597
| 703.5968
|  
|  
|}
|}
<nowiki/>* Besides the octave


[[Category:Garibaldi| ]] <!-- main article -->
[[Category:Garibaldi| ]] <!-- Main article -->
[[Category:Temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Schismatic family]]
[[Category:Schismatic family]]
[[Category:Marvel temperaments]]
[[Category:Marvel temperaments]]
[[Category:Gariboh clan]]
[[Category:Gariboh clan]]
[[Category:Hemifamity temperaments]]
[[Category:Hemifamity temperaments]]
Retrieved from "https://en.xen.wiki/w/Garibaldi"